Question: A goblet contains $4$ red marbles, $3$ green marbles, and $5$ blue marbles. If we choose a marble, then another marble without putting the first one back in the goblet, what is the probability that the first marble will be green and the second will be green as well? Write your answer as a simplified fraction.
Solution: The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened In this case, event A is picking a green marble and leaving it out. Event B is picking another green marble. Let's take the events one at at time. What is the probability that the first marble chosen will be green? There are $3$ green marbles, and $12$ total, so the probability we will pick a green marble is $\dfrac{3} {12}$ After we take out the first marble, we don't put it back in, so there are only $11$ marbles left. Also, we've taken out one of the green marbles, so there are only $2$ left altogether. So, the probability of picking another green marble after taking out a green marble is $\dfrac{2} {11}$ Therefore, the probability of picking a green marble, then another green marble is $\dfrac{3}{12} \cdot \dfrac{2}{11} = \dfrac{1}{22}$